A complex number is said to be an algebraic number if it is algebraic over Q and an algebraic integer if it is the root of a manic polynomial in Z[xJ.

(a) If u is an algebraic number, there exists an integer n such that nu is an algebraic integer.

(b) If r ϵ Q is an algebraic integer, then r ϵ Z.

(c) If u is an algebraic integer and n ϵ Z, then u + n and nu are algebraic integers.

(d) The sum and product of two algebraic integers are algebraic integers.